Total-Lagrangian vectorial lattice Boltzmann method for finite-strain hyperelasticity with curved boundaries

Abstract

Finite-strain hyperelasticity on curved embedded domains poses a geometric challenge for lattice Boltzmann methods. After streaming across an embedded material surface, the missing population is recovered at the physical cut-link point, where the lattice direction, surface normal, and tangential deformation directions are generally distinct. We develop a total-Lagrangian vectorial lattice Boltzmann method that resolves this geometric mismatch for two- and three-dimensional hyperelastic dynamics. The continuum equations are written as a conservative first-order system for material velocity and deformation gradient. Vector-valued populations are chosen so that their moments recover the state and the material-coordinate Piola fluxes, giving D2Q4\(\times\)6 and D3Q6\(\times\)12 schemes from one \(D\)-dimensional construction. Curved boundaries are embedded by a level set and closed link by link through opposite-population moment identities, cut-link interpolation, and local geometric information at the boundary point. The reconstruction is coupled to a compatibility projection that keeps the recovered displacement aligned with the evolved deformation gradient on embedded active-node graphs. The resulting method extends the previous grid-aligned two-dimensional formulation to curved domains and three-dimensional lattices while retaining explicit collide-stream updates on Cartesian grids. Benchmarks in two and three dimensions show agreement with exact finite-strain fields, nonlinear radial boundary-value problems, and finite-element references.